Abstract: In this talk we present a CUR matrix approximation that uses a novel convex optimization formulation to select the columns and rows of the data matrix for inclusion in C and R, respectively. We discuss implementation of the algorithm using the surrogate functional of Daubechies et al. [Communications on Pure and Applied Mathematics, 57.11 (2004)] and extend the theoretical guarantees of this approach to our formulation. Applications using CUR as a feature selection method for classification will be shown, if time. In addition, the proximal operator of the L-infinity norm is used in our CUR algorithm. We present a neural network approximation to this proximal operator that uses a novel feature selection process based on moments of the input data in order to allow vectors of varying lengths to be input into the network.
Abstract: Explicit descriptions of low degree K3 surfaces lead to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these two compactifications was studied by Shah and Looijenga, and revisited in the work of Laza and O’Grady. This latter work also provided a conjectural description for the case of degree four K3 surfaces. I will survey these results, and discuss a verification of this conjectural picture using tools from K-moduli. This is based on joint work with Kristin DeVleming and Yuchen Liu.
Abstract: There has been extensive study of diagonalization of matrices. Diagonalization can be viewed as using a similarity transform to concentrate the magnitude of all entries within as small a subset of entries as possible. We present results in our talk on what can be viewed as reversing this process, namely spreading out the magnitudes of entries as uniformly as possible.
Abstract:Â Real-world financial data can be multimodal distributed, and generating multimodal distributed real-world data has become a challenge to existing generative adversarial networks (GANs). For example, neural stochastic differential equations (Neural SDEs), treated as infinite-dimensional GANs, are only capable of generating unimodal time series data. In this talk, we present a novel time series generator, named directed chain GANs (DC-GANs), which inserts a time series dataset (called a neighborhood process of the directed chain or input) into the drift and diffusion coefficients of the directed chain SDEs with distributional constraints. DC-GANs can generate new time series of the same distribution as the neighborhood process, and the neighborhood process will provide the key step in learning and generating multimodal distributed time series. Signature from rough path theory will be used to construct the discriminator. Numerical experiments on financial data are presented and show a consistent outperformance over state-of-the-art benchmarks with respect to measures of distribution, data similarity, and predictive ability. If time permits, I will also talk about using Signature to solve mean-field games with common noise.
Abstract:Â The curve graph $\mathcal{C}$ of a finite genus surface $\Sigma_g$ is a central object in the study of the mapping class group of $\Sigma_g$. It exhibits many remarkable combinatorial properties. One, despite being an infinite, locally infinite graph Bestvina, Bromberg and Fujiwara proved that it has finite chromatic number. Two, its structure can be probed by finite rigid subgraphs: subgraphs $X \subset \mathcal{C}$ such that any locally injective map $X \to \mathcal{C}$ is the restriction of a global automorphism of $\mathcal{C}$. The outer automorphism groups of finite-rank free groups are studied by analogy with mapping class groups, though the analogy is imperfect. One analog of the curve graph is the sphere graph of a connect sum of $S^1\times S^2$s. In this talk I will introduce the sphere graph and discuss recent work investigating analogous combinatorial structure: an upper bound on the chromatic number (joint with SJSU students B. Haffner, E. Ortiz, and O. Sanchez) and the construction of finite rigid subgraphs (joint with C. Leininger).
Abstract: Cryptographic protocols with computational security are those that obtain security by restricting adversaries to only perform efficient actions. In the quantum setting, computational assumptions have been used to construct secure quantum protocols that utilize only classical communication. In this talk, I will focus on a primitive known as Trapdoor Claw-Free (TCF) Functions. TCFs have been used to construct many quantum protocols that only utilize classical communication. I will discuss their construction and explain how their properties can be used to obtain security against quantum adversaries.
Abstract: Mosquito-borne diseases endemic to areas with tropical climates have been spreading in temperate regions of the world with greater frequency in recent years. Numerous factors contribute to this spread, including urbanization; increases in global travel; and changes in temperature, precipitation, and humidity patterns due to climate change. Mathematical modeling is a useful tool to examine how these different influences impact transmission and spread of arboviruses and for projecting how potential future changes in these factors could affect arbovirus dynamics. Models have been employed for years to study disease dynamics, but diseases emerging in new regions present particular challenges. Here, we discuss models developed to study the introduction, emergence, and spread of dengue fever in Argentina. Dengue, caused by a virus transmitted by Aedes aegypti mosquitoes, first emerged in temperate Argentinian cities in 2009, and multiple outbreaks of increasing incidence have occurred since. With particular focus on the role of climate in dengue emergence, we present mathematical models designed to study meteorological influences on seasonal Aedes aegypti and dengue dynamics in temperate Argentinian cities, and we show how different seasonal patterns influence the risk of outbreaks. We also investigate potential influences of climate change on risk of dengue transmission in the future. We discuss the implications of our work on dengue and mosquito mitigation strategies, and we address some of the issues and areas for improvement in modeling emerging arboviruses.
Abstract: It is well known that many of the standard integral formulations for time-harmonic electromagnetic scattering break down, both numerically and analytically, as the frequency tends to zero. In many instances, the breakdown is directly due to the use of electric current as the fundamental unknown (as opposed to both current and charge, or other non-physical variables). It is somewhat less well known that similar instabilities arise for scattering problems in perfectly conducting half-spaces or piecewise layered media. In this talk we show a direct extension of the classic Lorenz-Debye-Mie scattering theory from spheres to half-spaces and layered media, and introduce a 'generalized Debye' formulation which is immune from low-frequency breakdown and gracefully decouples the non-physical unknowns in the limit. Each of these formulations is based on the classic Sommerfeld formulation of half-space scattering, which is equivalent to transverse Fourier transforms of the underlying Green's function.
Abstract: Convex functions of Gaussian vectors are prominent objectives in many fields of mathematical studies. In this talk, I will establish a new convexity for the logarithmic moment generating function for this object and draw two consequences. The first leads to the Paouris-Valettas small deviation inequality that arises from the study of convex geometry. The second provides a quantitative bound for the Dotsenko-Franz-Mezard conjecture in the Sherrington-Kirkpatrick mean-field spin glass model, which states that the logarithmic anneal partition function of negative replica is asymptotically equal to the free energy.
Abstract:Â A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric. And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful.
And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.
And what makes it all interesting is (3) applications. These turn out to include the proof of an old conjecture of Bondal and Van den Bergh about strong generation, a representability theorem that leads to a short, sweet proof of Serre's GAGA theorem, a proof of a conjecture by Antieau, Gepner and Heller about the non-existence of bounded t-structures on the category of perfect complexes over a singular scheme, as well as (most recently) a vast generalization and major improvement on an old theorem of Rickard's.
Abstract: It became clear recently that some version of the measure rigidity phenomenon as seen in the theorems of Ratner and Benoist-Quint occurs in a much more general setting. I will state some theorems and conjectures. I will also begin to discuss the connection to the dynamics of  a single hyperbolic diffeomorphism or flow, in particular to the relation between u-Gibbs states and SRB measures. (Talk runs 2:00 to 3:50)
Abstract: A knot K in $S^3$ is slice if it bounds a smooth disk in the four-ball; and is ribbon if this disk can be chosen to have no local maxima with respect to the radial height function on B^4. An old question of Fox asks if every slice knot is ribbon.
Let K be a knot equipped with a dihedral quotient of \pi_1(S^3\K). I'll explain how to extract an invariant of K from this data, using the signature of a certain 4-manifold which is a branched cover of B^4 over a properly embedded surface F with boundary K. I'll describe how this invariant can obstruct K from bounding, in alternate cases, either ribbon or slice disks in the four-ball. And, I'll give a necessary and sufficient condition for the existence of such a surface.
Abstract: A knot K in $S^3$ is slice if it bounds a smooth disk in the four-ball; and is ribbon if this disk can be chosen to have no local maxima with respect to the radial height function on B^4. An old question of Fox asks if every slice knot is ribbon.
Let K be a knot equipped with a dihedral quotient of \pi_1(S^3\K). I'll explain how to extract an invariant of K from this data, using the signature of a certain 4-manifold which is a branched cover of B^4 over a properly embedded surface F with boundary K. I'll describe how this invariant can obstruct K from bounding, in alternate cases, either ribbon or slice disks in the four-ball. And, I'll give a necessary and sufficient condition for the existence of such a surface.